Fine Structures of Hyperbolic Diffeomorphisms

359 Pages·2009·3.52 MB·English

by  Alberto A. Pinto, David A. Rand, Flávio Ferreira (auth.)

Springer Monographs in Mathematics
Alberto A. Pinto • David A. Rand •
Flávio Ferreira
Fine Structures
of Hyperbolic
Diffeomorphisms
Alberto A. Pinto David A. Rand
University of Minho Mathematics Institute
Departamento de Matemática (DM) University of Warwick
Campus de Gualtar Coventry, CV4 7AL
4710 - 057 Braga UK
Portugal
In celebration of the 60th birthday of
David A. Rand
For
Maria Guiomar dos Santos Adrego Pinto
B¨arbel Finkensta¨dt and the Rand kids: Ben, Tamsin, Rupert and
Charlotte
Fernanda Am´elia Ferreira and Fla´vio Andr´e Ferreira
Family and friends
Dedicated to Dennis Sullivan and Christopher Zeeman.
VI
Acknowledgments
Dennis Sullivan had numerous insightful discussions with us on this
work. In particular, we discussed the construction of solenoid functions,
train-tracks, self-renormalizable structures and pseudo-smooth structures for
pseudo-Anosov diffeomorphisms.
We would like to acknowledge the invaluable help and encouragement of
family, friends and colleagues, especially Abdelrahim Mousa, Alby Fisher,
Aldo Portela, Aloisio Arau´jo, Araga˜o de Carvalho, Athanasios Yannakopou-
los, Baltazar de Castro, B¨arbel Finkensta¨dt, Bruno Oliveira, Carlos Matheus,
Carlos Rocha, Charles Pugh, Dennis Sullivan, Diogo Pinheiro, Edson de Faria,
´
Enrique Pujals, Etienne Ghys, Fernanda Ferreira, Filomena Loureiro, Gabriela
Goes, Helena Ferreira, Henrique Oliveira, Hugo Sequeira, Humberto Mor-
eira, Isabel Labouriau, Jacob Palis, Joana Pinto, Joana Torres, Joa˜o Almeida,
Joaquim Baia˜o, John Hubbard, Jorge Buescu, Jorge Costa, Jos´e Gon¸calves,
Jos´e Martins, Krerley Oliveira, Lambros Boukas, Leandro Almeida, Leonel
Pias, Luciano Castro, Luis Magalha˜es, Luisa Magalh˜aes, Marcelo Viana,
Marco Martens, Maria Monteiro, Mark Pollicott, Marta Faias, Martin Peters,
Mauricio Peixoto, Miguel Ferreira, Mikhail Lyubich, Nelson Amoedo, Nico
Stollenwerk, Nigel Burroughs, Nils Tongring, Nuno Azevedo, Pedro Lago, Pa-
tricia Gonc¸alves, Robert MacKay, Rosa Esteves, Rui Gon¸calves, Saber Elaydi,
Sebastian van Strien, Sofia Barros, Sofia Cerqueira, Sousa Ramos, Stefano
Luzzatto, Stelios Xanthopolous, Telmo Parreira, Vilton Pinheiro, Warwick
Tucker, Welington de Melo, Yunping Jiang and Zaqueu Coelho.
We thank IHES, CUNY, SUNY, IMPA, the University of Warwick and
the University of Sa˜o Paulo for their hospitality. We also thank Calouste Gul-
benkian Foundation, PRODYN-ESF, Programs POCTI and POCI by FCT
and Minist´erio da Ciˆencia e da Tecnologia, CIM, Escola de Ciˆencias da Uni-
versidade do Minho, Escola Superior de Estudos Industriais e de Gesta˜o do
Instituto Polit´ecnico do Porto, Faculdade de Ciˆencias da Universidade do
Porto, Centros de Matema´tica da Universidade do Minho e da Universidade do
Porto, the Wolfson Foundation and the UK Engineering and Physical Sciences
Research Council for their financial support. We thank the Golden Medal dis-
tinction of the Town Hall of Espinho in Portugal to Alberto A. Pinto.
Alberto Pinto
David Rand
Fla´vio Ferreira
Preface
The study of hyperbolic systems is a core theme of modern dynamics. On
surfaces the theory of the fine scale structure of hyperbolic invariant sets and
their measures can be described in a very complete and elegant way, and is
the subject of this book, largely self-contained, rigorously and clearly written.
It covers the most important aspects of the subject and is based on several
scientific works of the leading research workers in this field.
This book fills a gap in the literature of dynamics. We highly recommend
it for any Ph.D student interested in this area. The authors are well-known
experts in smooth dynamical systems and ergodic theory.
Now we give a more detailed description of the contents:
Chapter 1. The Introduction is a description of the main concepts in hyper-
bolic dynamics that are used throughout the book. These are due to Bowen,
Hirsch, Man˜´e, Palis, Pugh, Ruelle, Shub, Sinai, Smale and others. Stable and
r
unstable manifolds are shown to be C foliated. This result is very useful in a
number of contexts. The existence of smooth orthogonal charts is also proved.
This chapter includes proofs of extensions to hyperbolic diffeomorphisms of
some results of Man˜´e for Anosov maps.
Chapter 2. All the smooth conjugacy classes of a given topological model
are classified using Pinto’s and Rand’s HR structures. The affine structures of
Ghys and Sullivan on stable and unstable leaves of Anosov diffeomorphisms
are generalized.
Chapter 3. A pair of stable and unstable solenoid functions is associated
to each HR structure. These pairs form a moduli space with good topologi-
cal properties which are easily described. The scaling and solenoid functions
introduced by Cui, Feigenbaum, Fisher, Gardiner, Jiang, Pinto, Quas, Rand
and Sullivan, give a deeper understanding of the smooth structures of one and
two dimensional dynamical systems.
Chapter 4. The concept of self-renormalizable structures is introduced.
With this concept one can prove an equivalence between two-dimensional hy-
perbolic sets and pairs of one-dimensional dynamical systems that are renor-
1+
malizable (see also Chapter 12). Two C hyperbolic diffeomorphisms that
VIII Preface
are smoothly conjugate at a point are shown to be smoothly conjugate. This
extends some results of de Faria and Sullivan from one-dimensional dynamics
to two-dimensional dynamics.
Chapter 5. A rigidity result is proved: if the holonomies are smooth enough,
then the hyperbolic diffeomorphism is smoothly conjugate to an affine model.
This chapter extends to hyperbolic diffeomorphisms some of the results of
Avez, Flaminio, Ghys, Hurder and Katok for Anosov diffeomorphisms.
Chapter 6. An elementary proof is given for the existence and uniqueness of
Gibbs states for H¨older weight systems following pioneering works of Bowen,
Paterson, Ruelle, Sinai and Sullivan.
Chapter 7. The measure scaling functions that correspond to the Gibbs
measure potentials are introduced.
Chapter refsmeasures. Measure solenoid and measure ratio functions are
1+
introduced. They determine which Gibbs measures are realizable by C hy-
1+
perbolic diffeomorphisms and by C self-renormalizable structures.
1+
Chapter 9. The cocycle-gap pairs that allow the construction of all C
hyperbolic diffeomorphisms realizing a Gibbs measure are introduced.
Chapter 10. A geometric measure for hyperbolic dynamical systems is
defined. The explicit construction of all hyperbolic diffeomorphisms with such
a geometric measure is described, using the cocycle-gap pairs. The results of
this chapter are related to Cawley’s cohomology classes on the torus.
Chapter 11. An eigenvalue formula for hyperbolic sets on surfaces with
an invariant measure absolutely continuous with respect to the Hausdorff
measure is proved. This extends to hyperbolic diffeomorphisms the Livˇsic-
Sinai eigenvalue formula for Anosov diffeomorphisms preserving a measure
absolutely continuous with respect to Lebesgue measure. Also given here is
an extension to hyperbolic diffeomorphisms of the results of De la Llave, Marco
and Moriyon on the eigenvalues for Anosov diffeomorphisms.
1+
Chapter 12. A one-to-one correspondence is established between C arc
1+ 1+
exchange systems that are C fixed points of renormalization and C hyper-
bolic diffeomorphisms that admit an invariant measure absolutely continuous
with respect to the Hausdorff measure. This chapter is related to the work of
Ghys, Penner, Rozzy, Sullivan and Thurston. Further, there are connections
with the theorems of Arnold, Herman and Yoccoz on the rigidity of circle
diffeomorphisms and Denjoy’s Theorem. These connections are similar to the
ones between Harrison’s conjecture and the investigations of Kra, Norton and
Schmeling.
Chapter 13. Pinto’s golden tilings of the real line are constructed (see
Pinto’s and Sullivan’s d-adic tilings of the real line in the Appendix C). These
golden tilings are in one-to-one correspondence with smooth conjugacy classes
of golden diffeomorphisms of the circle that are fixed points of renormalization,
and also with smooth conjugacy classes of Anosov diffeomorphisms with an in-
variant measure absolutely continuous with respect to the Lebesgue measure.
The observation of Ghys and Sullivan that Anosov diffeomorphisms on the
Preface IX
torus determine circle diffeomorphisms having an associated renormalization
operator is used.
Chapter 14. Thurston’s pseudo-Anosov affine maps appear as periodic
points of the geodesic Teichmu¨ller flow. The works of Masur, Penner, Thurston
and Veech show a strong link between affine interval exchange maps and
pseudo-Anosov affine maps. Pinto’s and Rand’s pseudo-smooth structures
near the singularities are constructed so that the pseudo-Anosov maps are
smooth and have the property that the stable and unstable foliations are
uniformly contracted and expanded by the pseudo-Anosov dynamics. Classi-
cal results for hyperbolic dynamics such as Bochi-Man˜´e and Viana’s duality
extend to these pseudo-smooth structures. Blow-ups of these pseudo-Anosov
diffeomorphisms are related to Pujals’ non-uniformly hyperbolic diffeomor-
phisms.
Appendices. Various concepts and results of Pinto, Rand and Sullivan for
one-dimensional dynamics are extended to two-dimensions. Ratio and cross-
ratio distortions for diffeomorphisms of the real line are discussed, in the spirit
of de Melo and van Strien’s book.
Rio de Janeiro, Brazil Jacob Palis
July 2008 Enrique R. Pujals
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Stable and unstable leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Interval notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Basic holonomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Foliated atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
ι
1.7 Foliated atlas A (g, ρ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Straightened graph-like charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9 Orthogonal atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.10 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 HR structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 HR - H¨older ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Foliated atlas A(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 HR Orthogonal atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Complete invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.8 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Realized solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 H¨older continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Cylinder-gap condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7 Solenoid functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Further literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43